Calculating the Diagonal Length of a Cube

The question asks us to find the length of a diagonal of a cube given that its side length is 3 cm. A cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. When we talk about the diagonal of a cube, we usually mean the space diagonal, which connects two opposite vertices passing through the interior of the cube.

Understanding the Cube Diagonal Formula

To find the length of the space diagonal of a cube, we can use a specific formula derived from the Pythagorean theorem. If the side length of the cube is denoted by 'a', the length of the space diagonal, often denoted by 'd', can be calculated using the formula:

\[d = a\sqrt{3}\]

Let's understand how this formula is derived. Consider a cube with side length 'a'.

• First, consider one face of the cube. This face is a square with side length 'a'. The diagonal of this face (let's call it 'f') can be found using the Pythagorean theorem: \[f^2 = a^2 + a^2 = 2a^2\] So, the face diagonal is \[f = \sqrt{2a^2} = a\sqrt{2}\]
• Now, consider a right-angled triangle formed by one edge of the cube (length 'a'), the face diagonal 'f' (length \(a\sqrt{2}\)), and the space diagonal 'd'. This triangle is formed by connecting one vertex, the diagonally opposite vertex on the same face, and the vertex diagonally opposite through the cube's interior. The edge of length 'a' is perpendicular to the face containing the diagonal 'f'.
• Using the Pythagorean theorem on this triangle: \[d^2 = a^2 + f^2\] Substitute the value of f we found: \[d^2 = a^2 + (a\sqrt{2})^2\] \[d^2 = a^2 + 2a^2\] \[d^2 = 3a^2\]
• Taking the square root of both sides gives the formula for the space diagonal: \[d = \sqrt{3a^2} = a\sqrt{3}\]

Calculating the Diagonal Length for a 3 cm Side Cube

The problem states that the side length of the cube is 3 cm. So, we have \(a = 3\) cm.

Using the formula for the space diagonal:

\[d = a\sqrt{3}\]

Substitute the value of 'a':

\[d = 3\sqrt{3} \text{ cm}\]

Therefore, the length of the diagonal of the cube with a side length of 3 cm is \(3\sqrt{3}\) cm.

Revision Table: Cube Properties

Additional Information on Cube Geometry

Understanding the properties of a cube is fundamental in solid geometry. The cube is a special case of several geometric solids, including a rectangular prism, a square prism, and a parallelepiped. Its symmetry makes its calculations straightforward.

• All edges of a cube are equal in length.
• All faces are congruent squares.
• All angles between faces are 90 degrees.
• A cube has 12 edges, 8 vertices, and 6 faces.
• There are 4 space diagonals in a cube, and they all have the same length and intersect at the center of the cube.
• There are 12 face diagonals in a cube, 2 on each of the 6 faces.

Knowing the relationship between the side length, face diagonal, and space diagonal using the Pythagorean theorem is a key concept for solving problems involving cubes and other rectangular solids.